Mathematics for the interested outsider

Special kinds of morphisms, subobjects, and quotient objects

Last week I wasusing the word “invertible” as if it was perfectly clear. Well, it should be, more or less, since categories are pretty similar to monoids, but I should be a bit more explicit. First, though, there’s a few other kinds of morphisms we should know.

We want to motivate these definitions from what we know of sets, but the catch is that sets are actually pretty special. Some properties turn out to be the same when applied to sets, though they can be different in other categories.

First of all, let’s look at injective functions. Remember that these are functions where implies . That is, distinct inputs produce distinct outputs. Now we can build a function as follows: if for some we define . This is well-defined because at most one can work, by injectivity. Then for all the other elements of we just assign them to random elements of . Now the composition is the identity function on because for all . We say that the function has a (non-unique) “left inverse”.

Now since has a left inverse there’s something else that happens: if we have two functions and both from to , and if then . That is, is “left cancellable”.

Now in any category we say a morphism is a “monomorphism” (or “a mono”, or “ is monic”) if it is left cancellable, whether or not the cancellation comes from a left inverse as above. If has a left inverse we say is “injective” or that it is “an injection”. By the same argument as above, every injection is monic, but in general not all monos are injective. In the two concepts are the same.

Similarly, a surjective function has a right inverse , and is thus right cancellable. We say in general that a right cancellable morphism is an “epimorphism” (or “an epi”, or “ is epic”). If the right cancellation comes from a right inverse, we say that is “surjective”, or that it is “a surjection”. Again, every surjection is epic, but not all epis are surjective. In the two concepts are again the same.

If a morphism is both monic and epic then we call it a “bimorphism”, and it can be cancelled from either side. If it is both injective and surjective we call it an “isomorphism”. All isomorphisms are bimorphisms, but not all bimorphisms are isomorphisms. If is an isomorphism, then we can show (try it) that the left and right inverses are not only unique, but are the same, and we call the (left and right) inverse . When I said “invertible” last week I meant that such an inverse exists.

We’ve already seen these terms in other categories. In groups and rings we have monomorphisms and epimorphisms, which are monos and epis in the categories and .

Now recall that any subset of a set comes with an injective function “including” into . Similarly, subgroups and subrings come with “inclusion” monomorphisms. We generalize this concept and define a “subobject” of an object in a category to be a monomorphism . In the same way we generalize quotient groups and quotient rings by defining a “quotient objects” of to be epimorphisms .

Notice that we define a subobject to be an arrow, and we allow any monomorphism. Consider the function defined by and . It seems odd at first, but we say that this is a subobject of . The important thing here is that we don’t define these concepts in terms of elements of sets, but in terms of arrows and their relations to each other. We “can’t tell the difference” between and since they are isomorphic as sets. If we just look at the arrow and the usual inclusion arrow of , they pick out the same subset of so we may as well consider them to be the same subset.

Let’s be a little more general here. Let and be two subobjects of . We say that “factors through” if there is an arrow so that . If we take the class of all subobjects of (all monomorphisms into ) we can give it the structure of a preorder by saying if factors through . It should be straightforward to verify that this is a preorder.

Now we can turn this preorder into a partial order as usual by identifying any two subobjects which factor through each other. If and then . Since is monic we can cancel it from the left and find that . similarly we find that . That is, and are inverses of each other, and so and are isomorphic as subobjects of . Conversely, if and are isomorphic subobjects then and factor through each other by an isomorphism . This gives us a partial order on (equivalence classes of) subobjects of . If the class of equivalence classes of subobjects is in fact a proper set for every object we say that our category is “well-powered”.

The preceding two paragraphs can be restated in terms of quotient objects. Just switch the directions of all the arrows and the orders of all the compositions. We get a partial order on (equivalence classes of) quotient objects of . If the class of equivalence classes is a proper set for each object then we say that the category is “co-well-powered”.

It should be noted that even though isomorphic subobjects come with an isomorphism between their objects, just having an isomorphism between the objects is not enough. One toy example is given in the comments below. Another is to consider two distinct one-element subsets of a given set. Clearly the object for each is a singleton, and all singletons are isomorphic, but the two subsets are not isomorphic as subobjects.

As an exercise, consider the category of commutative rings with unit and determine the partial order on the set of quotient objects of .

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[…] a “dual” definition, which we get by reversing all the arrows like this. For example, monos and epis are dual notions, as are subobjects and quotient objects. Just write down one definition in terms […]

Sorry if I bother you with this old post, but I didn’t catch one thing!
You wrote: “I $S_1$ and $S_2$ are isomorphic, then $f_1$ and $f_2$ will factor through each other”.
Could you prove it? In particular, I’m sure it works if $f_1,f_2$ are injective, but if not…?

So there is nothing we can say if f_1,f_2 are monic but not injective? Even if they have the same domain, it seems that they might not factor through each other. (for example: take a category with two objects, two identities and two arrows from the first object to the second, these arrows are monic but they don’t factor)

You say “g\circ f is the identity function on X because g(f(x))=x for all x\in X.” but it seems to me this is only true if f and g are surjective (i.e. f and g are bijections). Alternatively, we construct an f’ whose codomain is the image of f, and whose domain is the preimage of g.

(also minor correction: “h_1 and h_2 both from Y to *X*” [not Y to Z]).

Thanks for the quick response.
Yes, sorry. What I said for is essentially how you defined .
What I was thinking is that: is not the identity on , it is just the identity on the subset of X that is the preimage of f. But I just realised: you are assuming is total here, thus the preimage of f is. Apologies- I have a proclivity for thinking of functions as partial (too much functional programming!)

In Set it’s not quite true that monics coincide with functions that have a left inverse; consider what happens when the domain is empty.

You have to be on your guard when you use the term “quotient” for a class of epis. In the category of commutative rings, it happens that the inclusion of the integers in the rationals, for instance, is an epi, but normally one does not consider that the rationals form a quotient ring of the integers (people usually reserve “quotient ring” for a surjective homomorphism ). I’m not saying it’s wrong to define a quotient object in a category as a class of epis (people do that), but watch out about speaking of “quotient objects” in the category of rings in the same breath as “quotient rings” — there is a bad clash of terminology there.

I’ve never seen the terms “injective” and “surjective” used to describe morphisms in a general category which have left, resp., right inverses. This would definitely clash with standard usage; consider what people mean when they say “surjective continuous map”. Not all surjective continuous maps have sections. (By the way, some standard terms to introduce are “section” for a right inverse, and “retraction” for a left inverse.) Similarly for surjective homomorphisms in the category of groups.

Finally, not a nit but an observation: it’s probably a good idea to point out here that the statement that all epis in Set have a right inverse (a section) is equivalent to the axiom of choice.

Todd, a lot of the problem here is that there are at least three different naming conventions going on here. If I recall, when I wrote this I was taking Herrlich and Strecker as my source. And I specifically avoided “sect” and “retract” because those really hinge on certain topological references I’m not ready to introduce yet.

“Section” and “retraction” do not hinge on topological references. They are perfectly standard categorical terms. For example, “section” is used all the time when referring to splitting an exact sequence of groups (among other things).

I’ll check Herrlich and Strecker — that’s interesting if true. But I think you will agree that what people mean by surjective homomorphism or surjective continuous map doesn’t match the usage given here, and that it would be a good idea to point that out.

Please don’t be on the defensive — normally when I write, I do so in a spirit of trying to be of help!

Yes, the terms are used on their own, but you surely can’t deny that they’re meant to suggest the topological meanings. A section would never have been called a section without the idea of a section of a bundle, and a retraction would never have been called a retraction without the idea of a topological retraction. Why introduce a term when I can’t give some sort of motivation behind the name?

I’ll agree that section and retraction were (very probably, but I’m not an expert on the history) originally used in a topological context, but since then the meanings have expanded, and the way they are used today does not presuppose or hinge on topology. Anyway, I offered those terms as some useful current parlance, but if you don’t want to use them, that’s fine. They’re there for others to use if they want.

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